##
**Integral moments of \(L\)-functions.**
*(English)*
Zbl 1075.11058

Moments of \(L\)-functions are a central topic in analytic number theory. This paper brings forth a conjecture (via characteristic polynomials from Random Matrix Theory (RMT)) for the complete main term in the asymptotic formulas for a wide class of \(L\)-functions. In all cases when such an asymptotic formula was rigorously proved, the main term in question coincided with the expression predicted by the authors, which renders the present work quite important. Heretofore there have been several conjectures for particular \(L\)-functions, to mention here just the work of J. P. Keating and N. C. Snaith [Commun. Math. Phys. 214, 57–89 (2000; Zbl 1051.11047)] on the conjectural formula for the even integral moments of the Riemann zeta-function \(\zeta(s)\) on the “critical line” \(\operatorname{Re} s = 1/2\). The authors first introduce their class of \(L\)-functions (which is the slightly modified well-known Selberg class, for which the reader can see the review article of A. Kaczorowski and A. Perelli [Number Theory in Progress, Proc. Conf. in honour of A. Schinzel, de Gruyter, Berlin, 953–992 (1999; Zbl 0929.11028)]. To such an \(L\)-function
\[
L(s) = \sum_{n=1}^\infty a_n n^{-s} \qquad(\operatorname{Re} s > 1,\,a_n \ll_\varepsilon n^\varepsilon) \tag{1}
\]
in the region of absolute convergence they associate the characteristic polynomial of an \(N\times N\) unitary matrix \(A\), namely
\[
\Lambda_A(s) = \prod_{n=1}^N(1- s\text{ e}^{-i\theta_n}) = \sum_{n=0}^N a_ns^n, \tag{2}
\]
say, where the \(\theta_n\)’s are real. The analogy between (1) and (2), on the basis of RMT, is the starting point for the derivation of the authors’ moments conjecture. Before this is formulated, examples of primitive \(L\)-functions are given according to the families that are considered. Standard unitary, symplectic and orthogonal examples of families of \(L\)-functions are given.

The first main result is the conjecture that, for a fixed integer \(k\geq 1\) and a suitable non-negative smooth function \(g(t)\), as \(t\to\infty\), \[ \int_{-\infty}^\infty \biggl| \zeta \biggl({1\over2}+it\biggr)\biggr| ^{2k}g(t)\,dt = (1+o(1))\int_{-\infty}^\infty P_k\biggl(\log {t\over2\pi}\biggr)g(t)\,dt, \tag{3} \] where \(P_k\) is a polynomial of degree \(k^2\), whose coefficients depend on \(k\), that is given explicitly in an elegant, albeit complicated closed form. The authors actually conjecture, in all cases, an error term of the order \(O(T^{1/2+\varepsilon})\) for the moments over \([0,T]\), which in the general case this reviewer finds a bit optimistic. In any case, RMT gives no clue about the order of magnitude of the error term in the asymptotic formulas for moments of \(L\)-functions. The important cases \(k=1,2\), which are the only ones so far when the asymptotic formula in question was established, are discussed in detail. To assess the difficulty of the moments problem, note that (3) for all \(k\) implies the Lindelöf conjecture (that \(\zeta({1\over2}+it) \ll_\varepsilon | t| ^\varepsilon\)), while the converse is not known to hold. The moments conjecture does not seem to follow even from the notorious Riemann hypothesis (that all complex zeros of \(\zeta(s)\) have real part equal to 1/2), the most famous open problem in Mathematics.

The general “recipe” for conjecturing moments of \(L\)-functions is given in Section 4, by using the approximate functional equation and the functional equation for the \(L\)-function in question. This is done in such a way that those who will wish to work out conjectural formulas in specific examples will be able to do so following the authors’ instructions. There are many specific examples, remarks etc. which further enrich this far-reaching work. In addition, numerical calculations are given in Section 5. Coefficients of \(P_k(x)\) (see (3)) are given when \(2\leq k \leq 7\), and numerically computed moments are compared to the values obtained for the main term by the moments conjecture. In the ranges covered by the calculations, they are in good agreement. The moments conjecture set forth in this paper is a major event in the theory of \(L\)-functions which hopefully will incite research that will result in the proofs of the conjectured results.

The first main result is the conjecture that, for a fixed integer \(k\geq 1\) and a suitable non-negative smooth function \(g(t)\), as \(t\to\infty\), \[ \int_{-\infty}^\infty \biggl| \zeta \biggl({1\over2}+it\biggr)\biggr| ^{2k}g(t)\,dt = (1+o(1))\int_{-\infty}^\infty P_k\biggl(\log {t\over2\pi}\biggr)g(t)\,dt, \tag{3} \] where \(P_k\) is a polynomial of degree \(k^2\), whose coefficients depend on \(k\), that is given explicitly in an elegant, albeit complicated closed form. The authors actually conjecture, in all cases, an error term of the order \(O(T^{1/2+\varepsilon})\) for the moments over \([0,T]\), which in the general case this reviewer finds a bit optimistic. In any case, RMT gives no clue about the order of magnitude of the error term in the asymptotic formulas for moments of \(L\)-functions. The important cases \(k=1,2\), which are the only ones so far when the asymptotic formula in question was established, are discussed in detail. To assess the difficulty of the moments problem, note that (3) for all \(k\) implies the Lindelöf conjecture (that \(\zeta({1\over2}+it) \ll_\varepsilon | t| ^\varepsilon\)), while the converse is not known to hold. The moments conjecture does not seem to follow even from the notorious Riemann hypothesis (that all complex zeros of \(\zeta(s)\) have real part equal to 1/2), the most famous open problem in Mathematics.

The general “recipe” for conjecturing moments of \(L\)-functions is given in Section 4, by using the approximate functional equation and the functional equation for the \(L\)-function in question. This is done in such a way that those who will wish to work out conjectural formulas in specific examples will be able to do so following the authors’ instructions. There are many specific examples, remarks etc. which further enrich this far-reaching work. In addition, numerical calculations are given in Section 5. Coefficients of \(P_k(x)\) (see (3)) are given when \(2\leq k \leq 7\), and numerically computed moments are compared to the values obtained for the main term by the moments conjecture. In the ranges covered by the calculations, they are in good agreement. The moments conjecture set forth in this paper is a major event in the theory of \(L\)-functions which hopefully will incite research that will result in the proofs of the conjectured results.

Reviewer: Aleksandar Ivić (Beograd)

### MSC:

11M50 | Relations with random matrices |

11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |

15B52 | Random matrices (algebraic aspects) |